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United States Patent |
6,167,359
|
Demir
,   et al.
|
December 26, 2000
|
Method and apparatus for characterizing phase noise and timing jitter in
oscillators
Abstract
A nonlinear analysis for characterizing phase noise and timing jitter in
oscillators is disclosed. The method and apparatus utilize a nonlinear
differential equation to characterize the phase error of a given
oscillator. A precise stochastic characterization of timing jitter and
spectral dispersion is also disclosed based on the nonlinear differential
equation. Representative time and frequency domain computational
techniques are disclosed for characterizing the phase noise and timing
jitter of circuits. In addition, a single scalar constant, c, is utilized
to describe jitter and spectral spreading in a noisy oscillator.
Inventors:
|
Demir; Alper (Jersey City, NJ);
Mehrotra; Amit (El Cerrito, CA);
Roychowdhury; Jaijeet S. (Murray Hill, NJ)
|
Assignee:
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Lucent Technologies Inc. (Murray Hill, NJ)
|
Appl. No.:
|
096856 |
Filed:
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June 12, 1998 |
Current U.S. Class: |
702/191; 375/226; 375/227; 702/69; 702/111 |
Intern'l Class: |
H04B 015/00; H04B 017/00 |
Field of Search: |
702/190,191,69,70,76,111
703/2,3
375/226,227
|
References Cited
U.S. Patent Documents
3828138 | Aug., 1974 | Fletcher et al. | 375/344.
|
5943637 | Aug., 1999 | Okumura | 702/111.
|
6020782 | Feb., 2000 | Albert et al. | 327/552.
|
Other References
Alper Demir; "Phase Noise in Oscillators: DAES and Colored Noise Sources";
Bell Laboratories, pp. 170-177, 1998
Franz X. Kaertner; "Analysis of White and f.sup.-.alpha. Noise in
Oscillators"; International Journal of Circuit Theory of Application, vol.
18; pp. 485-519, 1990.
F.L. Walls et al., "Extending the Range and Accuracy of Phase Noise
Measurement" 42nd Annual Frequency Control Symposium; pp. 432-441, 1988.
Makiki Okumura and Hiroshi Tanimoto, A Time-Domain Method for Numerical
Noise Analysis of Oscillators, Proc. of the Asia and South Pacific Design
Automation Conference 477-82 (Jan. 1997).
|
Primary Examiner: Hoff; Marc S.
Assistant Examiner: Bui; Bryan
Attorney, Agent or Firm: Ryan, Mason & Lewis, LLP
Claims
We claim:
1. A method of characterizing the phase noise of an oscillator due to a
perturbation in said oscillator, said oscillator represented with a system
of differential equations, said method comprising the steps of:
estimating said perturbation;
deriving from said representation of said oscillator a nonlinear
differential equation representing said phase of said oscillator; and
using said perturbation as an input to said nonlinear differential equation
for said phase noise to characterize said phase noise.
2. The method according to claim 1, wherein said output of said oscillator
with phase noise is represented as a phase deviation to the unperturbed
oscillator response, x.sub.s (t+.alpha.(t)).
3. The method according to claim 2, wherein said nonlinear differential
equation is given by:
##EQU19##
4. The method according to claim 3, where v.sub.1.sup.T is calculated with
a time domain process.
5. The method according to claim 3, where v.sub.1.sup.T is calculated with
a frequency domain process.
6. A method of stochastically characterizing the phase noise of an
oscillator, said oscillator comprised of a plurality of component devices,
said oscillator represented with a system of differential equations, said
method comprising the steps of: estimating the random noise generated by
each of said component devices;
deriving from said representation of said oscillator a nonlinear
differential equation representing said phase of said oscillator; and
obtaining the stochastic characterization of said phase noise of said
oscillator based on a nonlinear differential equation for said phase
noise.
7. The method according to claim 6, wherein said output of said oscillator
with phase noise is represented as a phase deviation to the unperturbed
oscillator response, x.sub.s (t+.alpha.(t)).
8. The method according to claim 7, wherein said nonlinear differential
equation is given by:
##EQU20##
9. The method according to claim 8, where v.sub.1.sup.T is calculated with
a time domain process.
10. The method according to claim 8, where v.sub.1.sup.T is calculated with
a frequency domain process.
11. A method of stochastically characterizing the phase noise of an
oscillator, said oscillator comprised of a plurality of component devices,
said method comprising the steps of: estimating the white random noise
generated by each of said component devices;
characterizing said phase noise of said oscillator with a single scalar
constant, said scalar constant being the rate of change of the linearly
increasing variance of said phase noise; and
characterizing the output of said oscillator with phase noise as a
stationary stochastic process using a statistical ensemble average that is
not averaged over time.
12. The method according to claim 11, wherein said output of said
oscillator with phase noise is represented as a phase deviation to the
unperturbed oscillator response, x.sub.s (t+.alpha.(t)).
13. The method according to claim 12, wherein said nonlinear differential
equation is given by:
##EQU21##
14. The method according to claim 13, wherein v.sub.1.sup.T is calculated
with a time domain process.
15. The method according to claim 13, wherein v.sub.1.sup.T is calculated
with a frequency domain process.
16. The method according to claim 13, wherein said scalar constant, c, is
given by:
17. The method according to claim 13, further comprising the step of
characterizing the spectrum of said stationary process.
18. The method according to claim 17, wherein said spectrum, S(.function.),
is given by:
Description
FIELD OF THE INVENTION
The present invention relates to design tools for oscillators, and more
particularly, to a method and apparatus for characterizing phase noise and
timing jitter in oscillators.
BACKGROUND OF THE INVENTION
Oscillators are ubiquitous in physical systems, especially electronic and
optical systems, where oscillators are utilized in phase locked loops,
voltage controlled oscillators, microprocessors and transceivers. For
example, in radio frequency (RF) communication systems, oscillators are
used for frequency translation of information signals and for channel
selection. Oscillators are also present in digital electronic systems that
require a time reference (or clock signal) to synchronize operations.
Noise is a significant problem with oscillators, because introducing even
small noise into an oscillator leads to dramatic changes in its frequency
spectrum and timing properties. This phenomenon, peculiar to oscillators,
is known as phase noise or timing jitter. A perfect oscillator would have
localized tones at discrete frequencies (fundamental frequency, f.sub.0,
and harmonics, 2f.sub.0 and 3f.sub.0), as shown in FIG. 1A, but any
corrupting noise spreads these perfect tones, resulting in high power
levels at neighboring frequencies, as shown in FIG. 1B. Spreading is the
major contributor to undesired phenomena, such as interchannel
interference, leading to increased bit-error-rates (BER) in RF
communication systems.
Another manifestation of the same phenomenon is timing jitter. Timing
jitter is important in clocked and sampled-data systems. The uncertainties
in switching instants caused by noise lead to synchronization problems.
Thus, characterizing how noise affects oscillators is crucial for
practical applications. The problem, however, is challenging since
oscillators constitute a special class among noisy physical systems. The
autonomous nature of oscillators makes them unique in their response to
perturbations.
Considerable effort has been expended over the years in understanding phase
noise and in developing analytical, computational and experimental
techniques for its characterization. For a brief review of such
analytical, computational and experimental techniques, see, for example,
W. P. Robins, Phase Noise in Signal Sources, Peter Peregrinus (1991); B.
Razavi, Analysis, Modeling and Simulation of Phase Noise in Monothilic
Voltage-Controlled Oscillators, Proc. IEEE Custom Integrated Circuits
Conference (May 1995), each incorporated by reference herein.
In order to compensate for phase noise and timing jitter, the designers of
oscillators and devices incorporating oscillators utilize existing
commercial tools, such as the tools commercially available from Cadence
Design Systems and Hewlett Packard's EESOF, to analyze the single-side
band phase noise spectrum as a function of the offset from the fundamental
frequency. A typical single-side band phase noise spectrum is shown in
FIG. 2. Specific single-side band phase noise values are specified for a
given device at various frequencies, in a known manner. Conventional tools
for analyzing the single-side band phase noise spectrum typically assume
the peak power spectral density (PSD) 150, shown in FIG. 1B, is infinity.
Thus, the single-side band phase noise values 210 generated by
conventional tools likewise go to infinity, as shown in FIG. 2. It has
been found, however, that the estimate of the single-side band phase noise
actually has a shape as shown by the curve 220. As a result, if the device
being designed is analyzed at a frequency below a given offset frequency,
f.sub.off, 230, the oscillator will be over-designed to compensate for the
overestimated single-side band phase noise.
Despite the importance of the phase noise problem and the large number of
publications on the subject, a consistent and general treatment, and
computational techniques based on a sound theory, are still lacking. Thus,
a need exists for a method and apparatus using efficient numerical methods
for the characterization of phase noise. A further need exists for a
technique that is applicable to any oscillatory system, including
electrical systems, such as resonant, ring and relaxation oscillators, and
other systems, such as gravitational, optical, mechanical and biological
oscillators. Yet another need exists for a tool that estimates the actual
power spectral density (PSD) of a given oscillator and accurately
characterizes the oscillator's single-side band phase noise spectrum.
SUMMARY OF THE INVENTION
Generally, a nonlinear analysis for characterizing phase noise in
oscillators is disclosed. Computational methods in the time and frequency
domains are disclosed for accurately predicting the phase noise of an
oscillator. The disclosed method and apparatus utilize a nonlinear
differential equation that characterizes the phase error of a given
oscillator, which can be solved without approximations for random
perturbations. In addition, the present invention provides a precise
stochastic characterization of timing jitter and spectral dispersion.
In accordance with another aspect of the invention, representative time and
frequency domain computational techniques are disclosed for characterizing
the phase noise and timing jitter of circuits. The computational
techniques require only a knowledge of the steady state of the unperturbed
oscillator, and the values of the noise generators. In addition, the
separate contributions of noise sources, and the sensitivity of phase
noise to individual circuit devices and nodes, can be obtained.
In addition, in accordance with the invention, a single scalar constant, c,
is utilized to describe jitter and spectral spreading in a noisy
oscillator.
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1A illustrates the power spectral density (PSD) of an ideal oscillator
having localized tones at discrete frequencies;
FIG. 1B illustrates the power spectral density (PSD) of an oscillator in
the presence of phase noise;
FIG. 2 illustrates the single-side band phase noise characterization
produced by conventional tools, as well as in accordance with the present
invention;
FIG. 3 is a schematic block diagram illustrating a designer's workstation
of FIG. 1;
FIG. 4A is a schematic block diagram illustrating a conventional
oscillator;
FIG. 4B illustrates the periodic signal, x.sub.s (t), produced by the
oscillator of FIG. 4A;
FIG. 4C illustrates the phase deviation, .alpha.(t), and orbital deviation,
y(t), in the periodic output of the unperturbed oscillator of FIG. 4A;
FIG. 5 is a flow chart describing the phase noise and timing jitter
characterization process of FIG. 3;
FIG. 6 is a schematic block diagram illustrating a conventional oscillator;
and
FIGS. 7A through 7F show various characterizations of the output of the
oscillator shown in FIG. 6.
DETAILED DESCRIPTION
As discussed herein, the present invention utilizes a nonlinear analysis to
characterize phase noise and timing jitter in oscillators. A nonlinear
differential equation is disclosed to characterize the phase error and
timing jitter of a given oscillator. In addition, the present invention
provides a precise stochastic characterization (based on the nonlinear
differential equation) of timing jitter and spectral dispersion.
FIG. 3 is a block diagram showing the architecture of an illustrative
designer workstation 300. In order to characterize the phase noise and
timing jitter in accordance with the present invention, the workstation
300, or other computing device, utilized by the designer, programmer or
developer, should contain a number of software components and tools. The
workstation 300 preferably includes a processor 310 and related memory,
such as a data storage device 320, which may be distributed or local. The
processor 310 may be embodied as a single processor, or a number of local
or distributed processors operating in parallel. The data storage device
320 and/or a read only memory (ROM) are operable to store one or more
instructions, which the processor 310 is operable to retrieve, interpret
and execute.
The data storage device 320 preferably includes one or more component
device specification file(s) 330, containing specifications on the noise,
b(t), contributed by each of the various component devices included in a
given oscillator. In addition, as discussed further below in conjunction
with FIG. 5, the data storage device 320 also includes a phase noise and
timing jitter characterization process 500. The phase noise and timing
jitter characterization process 500 disclosed herein is merely
representative of the processes which may be implemented to characterize
the phase noise and timing jitter in accordance with the present
invention. While the phase noise and timing jitter characterization
process 500 shown in FIG. 5 is based on a time domain technique, an
alternative process based on a frequency domain technique is discussed
below in a section entitled Frequency Domain Technique for Calculating
v.sub.1 (t).
NONLINEAR CHARACTERIZATION OF PHASE SHIFT AND JITTER
FIG. 4A illustrates an oscillator 400, consisting of a lossy inductor 410
and capacitor 420 circuit with an amplitude-dependent gain provided by a
nonlinear resistor 430. The nonlinear resistor 430 has a negative
resistance region which pumps energy into the circuit 400 when the voltage
across the capacitor 420 drops, thus maintaining stable oscillation, in a
known manner. A current source 450 is also present, representing external
perturbations due to noise. When there is no perturbation, in other words,
when b(t) is zero, the oscillator oscillates with a perfectly periodic
signal x.sub.s (t) 460, shown in FIG. 4B. The periodic signal x.sub.s (t)
is a vector consisting of the capacitor voltage and the inductor current
460, 470. In the frequency domain, the unperturbed waveform consists of a
series of impulses at the fundamental and harmonics of the time period, as
shown in FIG. 1A.
In general, the dynamics of any oscillator, such as the oscillator 400, can
be described by a system of differential equations:
x=.function.(x) Eq.1
where
x.epsilon.IR.sup.n and .function.(.multidot.):IR.sup.n .fwdarw.IR.sup.n.Eq.
2
For systems that have a periodic solution x.sub.s (t) (with period T) to
equation 1, there is a stable limit cycle in the n-dimensional solution
space. While equation 1 describes an oscillator 400 without perturbations
or noise, we are interested in the response of such oscillators 400 to a
small state-dependent perturbation of the form B(x)b(t), where
B(.multidot.):IR.sup.n .fwdarw.IR.sup.nxp and
b(.multidot.):IR.fwdarw.IR.sup.p. Eq.3
Thus, the perturbed system is described by
x=.function.(x)+B(x)b(t) Eq.4
The response of the oscillator 400 when b(t) is a known deterministic
signal, modifies the unperturbed periodic response, x.sub.s (t), of the
oscillator 400 to
x.sub.s (t+.alpha.(t))+y(t) Eq.5
where .alpha.(t) is a changing time shift, referred to herein as phase
deviation, in the periodic output of the unperturbed oscillator, and y(t)
is an additive component, referred to herein as orbital deviation, to the
phase-shifted oscillator waveform, as shown in FIG. 4C. .alpha.(t) and
y(t) can always be chosen such that .alpha.(t) will, in general, keep
increasing with time even if the perturbation b(t) is always small. The
orbital deviation, y(t), on the other hand, will always remain small.
As discussed below, the present invention provides equations for .alpha.(t)
and y(t) that lead to qualitatively different results about phase noise
compared to previous attempts. The equations for .alpha.(t) and y(t) of
the present invention are based on a nonlinear perturbation analysis that
is valid for oscillators, in contrast to previous approaches that rely on
linearization.
Thus, in accordance with the present invention, the nonlinear differential
equation for the phase shift .alpha.(t) is as follows:
##EQU1##
where v.sub.1 (t) is a periodically time-varying vector, referred to herein
as the Floquet vector. The derivations and proofs of the above results are
based on the Floquet theory, described in M. Farkas, Periodic Motions,
Springer-Verlag (1994), of linear periodically time-varying systems. From
equation 6, if the perturbation is orthogonal to the Floquet vector for
all t, the phase error .alpha.(t) is zero. The Floquet vector, in general,
has no relationship to the tangent vector x.sub.s (t) to the limit cycle.
Although some prior theories have assumed that if the perturbation is
orthogonal to the tangent vector, then there is no phase error, it has
been found that the direction of the perturbation that results in zero
phase error is the direction that is orthogonal to the Floquet vector
(generally, oblique to the tangent).
STOCHASTIC CHARACTERIZATION
When the perturbation b(t) is assumed to be random noise (a vector of white
noise processes), zero-crossing jitter and spectral purity (spreading of
the power spectrum) may be determined. Jitter and spectral spreading are
in fact closely related, and both are determined by the manner in which
.alpha.(t), now also a random process, spreads with time. As discussed
further below, it has been found that:
(i) the average spread of the jitter (mean-square jitter, or variance)
increases precisely linearly with time, such that,
E[.alpha..sup.2 (t)]=.sigma..sup.2 (t)=ct Eq.7
where:
##EQU2##
(ii) the power spectrum of the perturbed oscillator is a Lorentzian about
each harmonic (a Lorentzian is the shape of the squared magnitude of a
one-pole lowpass filter transfer function);
(iii) a single scalar constant, c, is sufficient to describe jitter and
spectral spreading in a noisy oscillator; and
(iv) the oscillator's output with phase noise (x.sub.s (t+.alpha.(t)) is a
stationary stochastic process.
Thus, if we define X.sub.1 to be the Fourier coefficients of x.sub.s (t):
##EQU3##
then, the spectrum of the stationary oscillator output (x.sub.s
(t+.alpha.(t)) is given by:
##EQU4##
where f.sub.0 =1/T is the fundamental frequency. The phase deviation
.alpha.(t) does not change the total power in the periodic signal x.sub.s
(t), but it alters the power density in frequency, i.e., the power
spectral density. As previously indicated, for the perfect periodic signal
x.sub.s (t), the power spectral density (PSD) has delta functions located
at discrete frequencies (the fundamental frequency and the harmonics), as
shown in FIG. 1A. The phase deviation .alpha.(t) spreads the power in
these delta functions in the form given in equation 10, which can be
experimentally observed with a spectrum analyzer, as shown in FIG. 1B. In
this manner, the finite values of the power spectral density (PSD) at the
carrier frequency and its harmonics can be obtained for a given oscillator
with the present invention (as opposed to prior art systems that predict
infinite noise power density at the carrier, as well as infinite total
integrated power). That the oscillator output is stationary reflects the
fundamental fact that noisy autonomous systems cannot provide a perfect
time reference.
In accordance with further features of the present invention, time and
frequency domain computational techniques efficient for practical circuits
are disclosed. The techniques of the present invention require only a
knowledge of the steady state of the unperturbed oscillator 400, and the
values of the noise generators. In addition, large circuits are handled
efficiently (computation/memory scale linearly with circuit size).
Furthermore, the separate contributions of noise sources, and the
sensitivity of phase noise to individual circuit devices and nodes, can be
obtained easily.
QUANTIFICATION OF JITTER AND SPECTRAL PROPERTIES
As discussed below, several popular characterizations of phase noise that
are used in the design of electronic oscillators can be obtained from the
stochastic characterization provided in the previous section.
i. Single-Sided Spectral Density and Total Power
The single-sided spectral density is given above by equation 10. The total
power, P.sub.tot, (the integral of the PSD over the range of defined
frequencies) is given by:
##EQU5##
It is noted that the total power in the periodic signal x.sub.s (t)
(without phase noise) is also equal to the expression in equation 11
(excluding the power in the DC part).
It is further noted that the phase deviation .alpha.(t) does not change the
total power in the periodic signal x.sub.s (t), but it alters the power
density in frequency, i.e., the power spectral density. As previously
indicated, for the perfect periodic signal x.sub.s (t), the power spectral
density has delta functions located at discrete frequencies (i.e., the
harmonics). The phase deviation alpha spreads the power in these delta
functions in the form given by equation 10, which can be experimentally
observed with a spectrum analyzer.
ii. Spectrum in dBm/Hz
For electrical oscillators, the state variable in the oscillator that is
observed as the output is usually a voltage or a current. The spectrum in
equation 10 is expressed as a function of frequency (.function. in Hz),
then the PSD is in units of volts.sup.2 /Hz and amps.sup.2 /Hz for a
voltage and a current state variable respectively. Then, the spectral
density of the expected (i.e., average, assuming that the stochastic
process (x.sub.s (t+.alpha.(t)) is ergodic) power dissipated in a 1
.OMEGA. resistor (with the voltage (current) output of the oscillator as
the voltage across (current through) the resistor) is equal to the PSD in
equation 10 (in watts/Hz), which is usually expressed in dBw/Hz as defined
by:
S.sub.dBw (.function.)=10 log.sub.10 (S(.function.) in watts/Hz)Eq.12
If S(.function.) is in miliwatts/Hz, then the PSD in dBm/Hz is given by:
S.sub.dBm (.function.)=10 log.sub.10 (S(.function.) in milliwatts/Hz)Eq.13
iii. Single-sideband Phase Noise Spectrum in dBc/Hz
In practice, designers are usually interested in the PSD around the first
harmonic, i.e., S(.function.) for .function. around .function..sub.0. The
single-sideband phase noise L(.function..sub.m) (in dBc/Hz) that is very
widely used in practice is defined as:
##EQU6##
For "small" values of c, and for 0.ltoreq.f.sub.m <<f.sub.0, equation 14
can be approximated as:
##EQU7##
Furthermore, for .pi..function..sub.0.sup.2 c<<.function..sub.m
<<.function..sub.0, L(.function..sub.m) can be approximated by:
##EQU8##
It is noted that the approximation of L(.function..sub.m) in equation 16
does not apply as .function..sub.m .fwdarw.0. For
0.ltoreq..function..sub.m <.pi..function..sub.0.sup.2 c, equation 16 is
not accurate, in which case the approximation in equation 15 should be
used.
iv. Timing Jitter
In some applications, such as clock generation and recovery, a designer is
interested in a characterization of the phase/time deviation .alpha.(t)
itself rather than the spectrum of (x.sub.s (t+.alpha.(t)). In these
applications, an oscillator generates a square-wave like waveform to be
used as a clock.
The effect of the phase deviation .alpha.(t) on such a waveform is to
create jitter in the zero-crossing or transition times. According to
equation 7, .alpha.(t) (for an autonomous oscillator) becomes a random
variable with a linearly increasing variance. If one of the transitions
(i.e., edges) of a clock signal is used as a reference (i.e., trigger)
transition and it is synchronizes with t=0, and if the clock signal is
perfectly periodic, then the transitions occur exactly at t.sub.k =kT,
k=1,2, . . . where T is the period. For a clock signal with a phase
deviation .alpha.(t) that has a linearly increasing variance as above, the
timing of the k-th transition t.sub.k will have a variance (i.e.,
mean-square error) given by:
E[(t.sub.k -kT).sup.2 ]=ckT Eq.17
The spectral dispersion caused by .alpha.(t) in an oscillation signal can
be observed with a spectrum analyzer. Similarly, a designer can observe
the timing jitter caused by .alpha.(t) using a sampling oscilloscope.
McNeill experimentally observed the linearly increasing variance for the
timing of the transitions of a clock signal generated by an autonomous
oscillator, as predicted by our theory (see J. A. McNeill, Jitter in Ring
Oscillators, PhD thesis, Boston University (1994)). Moreover, c (in
sec.sup.2.Hz) in equation 17 exactly quantifies the rate of increase of
timing jitter with respect to a reference transition. Another useful
figure of interest is the cycle-to-cycle timing jitter, i.e., the timing
jitter in one clock cycle, which has a variance cT.
v. Noise Source Contributions
The scalar constant c appears in the characterizations discussed above. It
is given by equation 8, where B(.multidot.): IR.sup.n .fwdarw.IR.sup.n x p
represents the modulation of intensities of the noise sources with the
large-signal state. Equation 8 can be rewritten as:
##EQU9##
where p is the number of the noise sources, i.e., the column dimension of
B(x.sub.s (.multidot.)), and B.sub.i (.multidot.) is the i-th column of
B(x.sub.s (.multidot.)) which maps the i-th noise source to the equations
of the system. Hence,
##EQU10##
represents the contribution of the i-th noise source to c. Thus, the ratio
##EQU11##
can be used as a figure of merit representing the contribution of the i-th
noise source to phase noise/timing jitter. Note that the phase error
.alpha.(t) is described by a nonlinear differential equation where the
noise sources are the excitations. Hence, one can not use the
superposition principle to calculate the phase error arising from multiple
noise sources. On the other hand, the phase error variance, .sigma..sup.2
(t)=ct, is linearly related to the noise sources, i.e., the variance of
phase error since two noise sources is the summation of the variances due
to the noise sources considered separately.
vi. Phase Noise Sensitivity
The phase noise/timing jitter sensitivity of the k-th equation (i.e., node)
can be defined as:
##EQU12##
since e.sub.k represents a unit intensity noise source added to the k-th
equation (i.e., connected to the k-th node) in equation 1. The phase noise
sensitivity of nodes can provide useful information in search for novel
oscillator architectures with low phase noise.
PROCESSES
As previously indicated, the phase noise and timing jitter characterization
process 500, shown in FIG. 5, is representative of the processes which may
be implemented to characterize the phase noise and timing jitter in
accordance with the present invention. While the phase noise and timing
jitter characterization process 500 shown in FIG. 5 is based on a time
domain technique, an alternative process based on a frequency domain
technique is discussed below in a section entitled Frequency Domain
Technique for Calculating v.sub.1 (t).
As shown in FIG. 5, the phase noise and timing jitter characterization
process 500 initially computes the large signal periodic steady state
solution x.sub.s (t) for 0.ltoreq.t.ltoreq.T by numerically integrating
equation 1, during step 510. In one embodiment, the shooting method
technique disclosed in K. S. Kundert et al., Steady-State Methods for
Simulating Analog and Microwave Circuits, Academic Publishers (1990) is
utilized.
Thereafter, the phase noise and timing jitter characterization process 500
computes the state transition matrix .PHI.(T,0) by numerically
integrating:
Y=A(t)Y, Y(0)=I.sub.n from 0 to T during step 520, where the Jacobian
##EQU13##
is T-periodic.
It is noted that .PHI.(T, 0)=Y(T). u.sub.1 (0) is then computed during step
530 using:
U.sub.1 (0)=x.sub.s (0). It is further noted that u.sub.1 (0) is an
eigencevector of .PHI.(T, 0) corresponding to the eigenvalue 1.
An eigenvector of .PHI..sup.T (T,0) corresponding to the eigenvalue 1 is
computed during step 540 and then scaled so that
v.sub.1 (0).sup.T u.sub.1 (0)=1 is satisfied. It is noted that v.sub.1 (0)
is an eigenvector of .PHI..sup.T (T,0) corresponding to the eigenvalue 1.
The periodic vector, v.sub.1 (t), for 0.ltoreq.t.ltoreq.T is computed
during step 550 by numerically solving the adjoint system:
y=-A.sup.T (t)y using v.sub.1 (0)=v.sub.1 (T) as the initial condition.
Finally, c is computed during step 560 using the following equation:
##EQU14##
In implementing the above algorithm, one can increase the efficiency by
saving LU factored matrices that need to be calculated during step 520 and
reuse them in step 550. If the periodic steady-state x.sub.s (t) of the
oscillator is calculated using the shooting method, referenced above, in
step 510, then the state transition matrix .PHI.(T, 0) of the linear
time-varying system obtained by linearizing the nonlinear oscillator
circuit around the periodic steady-state is already available. It can be
shown that the Jacobian of the nonlinear system of equations that is
solved in the shooting method (using Newton's method, to calculate the
initial condition that results in the periodic steady-state solution) is
equal to .PHI.(T, 0)-1 (see T. J. Aprille and T. N. Trick, A Computer
Algorithm To Determine The Steady-State Response Of Nonlinear Oscillators,
IEEE Transactions on Circuit Theory, CT-19(4):354-360 (July, 1972); T. J.
Aprille and T. N. Trick, Steady-State Analysis of Nonlinear Circuits With
Periodic Inputs, Proceedings of the IEEE, 60(1):108-114 (January, 1972)).
Moreover, one can avoid calculating .PHI.(T, 0) explicitly, and use
iterative methods both for the shooting method, and at step 540 to
calculate the eigenvector of .PHI..sup.T (T, 0) that corresponds to the
eigenvalue 1 (see R. Telichevesky et al, Efficient Steady-State Analysis
Based On Matrix-Free Krylov-Subspace Methods, Proc. Design Automation
Conference (June, 1995)). For high-Q oscillators, the iterative methods
can run into problems, because .PHI.(T, 0) may have several other
eigenvalues which are close to 1. In our implementation in SPICE, we
explicitly calculate .PHI.(T, 0) and perform a full eigenvalue/eigenvector
calculation, which allows us to investigate the properties of the
state-transition matrix for various oscillator circuits. Even with a full
eigenvalue/eigenvector calculation for .PHI.(T, 0), the phase noise
characterization algorithm discussed above is still very efficient. The
phase noise characterization comes "almost" for free, once the periodic
steady-state solution x.sub.s (t) is computed.
The state transition matrix is given by:
.PHI.(t,s)=U(t)exp(D(t-s))V(s)
where U(t) is a T-periodic nonsingular matrix, V(t)=U.sup.-1 (t) and
diag[.mu..sub.1, . . . .mu..sub.n ], where .mu..sub.I are the Floquet
(characteristic) exponents. exp(.mu..sub.i T) are called characteristic
multipliers.
Frequency Domain Technique for Calculating v.sub.1 (t)
As previously indicated, the phase noise and timing jitter characterization
process 500, shown in FIG. 5, can alternatively be based on a frequency
domain technique. In a frequency domain embodiment, the matrices U.sub.i
and V.sub.i are defined to be the Fourier components of U(t) and V(t),
such that,
##EQU15##
The block-Toeplitz matrices U and V are defined as follows:
##EQU16##
U and V are both invertible, since
##EQU17##
and U(t) and V(t) are both bi-orthonormal. D.sub.k (.omega.) is defined
as:
##EQU18##
In addition, D.sub.k (.omega.) is singular for .omega.=k.omega..sub.0, if
the oscillator is asymptotically orbitally stable. The frequency domain
conversion matrix, H(.omega.) of the oscillator is related to U,
D(.omega.) and V by:
H(.omega.)=UD.sup.-1 (.omega.)V.
H.sup.-1 (0) is a singular matrix (with rank-deficiency one) and the null
space of its transpose is spanned by the Fourier components of v.sub.1
(t), such that, ker(H.sup.-T (0))=[1 0 . . . 0 ][ . . . V.sub.-2 V.sub.-1
V.sub.0 V.sub.1 V.sub.2 . . . ]=[ . . . V.sub.1.sup.T,.sub.-2
V.sub.1.sup.T,.sub.-1 V.sub.1.sup.T,.sub.0 V.sub.1.sup.T,.sub.1
V.sub.1.sup.T,.sub.2 . . . ]
where V.sub.1,i are the Fourier coefficients of kv.sub.1.sup.T (t), for
some nonzero scalar k, such that,
kv.sub.1.sup.T (t)=.SIGMA.V.sub.1.sup.T,.sub.i e.sup.jw.sbsp.0.sup.it
H.sup.-T (0)is simply the transpose of the Harmonic Balance Jacobian matrix
of the oscillator at solution. Its null space can be found efficiently
even for large circuits by using iterative linear algebra techniques. For
a more detailed discussion of such techniques, see J. Roychowdhury et al.,
Cyclostationary Noise Analysis of Large RF Circuits With Multi-Tone
Excitations, IEEE Journal of Solid-State Circuits, March, 1998,
incorporated by reference herein. Hence a scaled version of v.sub.1.sup.T
(t) can be found easily. The scaling constant k can be found by applying
v.sub.1.sup.T (t) u.sub.1 (t)=1, u.sub.1 (t) having first been obtained by
differentiating the large-signal steady state solution of the oscillator.
PRACTICAL EXAMPLE
The oscillator 600, shown in FIG. 6, consists of a Tow-Thomas second-order
bandpass filter and a comparator. If the OpAmps are considered to be
ideal, it can be shown that the oscillator 600 is equivalent (in the sense
of the differential equations that describe it) to a parallel RLC circuit
in parallel with a nonlinear voltage-controlled current source (or
equivalently a series RLC circuit in series with a nonlinear
current-controlled voltage source) as shown in FIG. 4A. For Q=1 and
f.sub.0 =6.66 kHz, we performed a phase noise characterization of this
oscillator using our numerical methods, and computed the periodic
oscillation waveform x.sub.s (t) for the output and c=7.56.times.10.sup.-8
sec.sup.2.Hz. FIG. 7A shows the PSD of the oscillator output computed
using equation 10, and FIG. 7B shows the spectrum analyzer measurement.
FIG. 7C shows a blown up version of the PSD around the first harmonic. The
single-sideband phase noise spectrum using both equations 14 and 15 is in
FIG. 7D. It is again noted that equation 15 cannot predict the PSD
accurately below the cut-off frequency .function..sub.c =.pi.f.sub.0.sup.2
c=10.56 Hz (marked with a * in FIG. 7D of the Lorentzian).
The oscillator model that was simulated has two state variables and a
single stationary noise source. FIG. 7E shows a plot of the periodic
nonnegative scalar
v.sub.1.sup.T (t)B(x.sub.s (t))B.sup.T (x.sub.s (t))v.sub.1
(t)=(v.sub.1.sup.T B).sup.2
where 2.times.1B is independent of t since the noise source is stationary.
Recall from equation 18 that c is the time average of this scalar that is
periodic in time.
c can also be obtained relatively accurately in this case using Monte-Carlo
analysis. We simulated the circuit 600 with 10000 random excitations and
averaged the results to obtain the mean-square difference between the
perturbed and unperturbed systems as a function of time. FIG. 7F
illustrates the result, the slope of the envelope of which determines c.
The Monte-Carlo simulations required small time-steps to produce accurate
results, since numerical integration methods easily lose accuracy for
autonomous circuits.
The total computation time for Monte-Carlo was about 10 hours on a fast SGI
workstation (R2000 CPU), whereas the present invention required about 20
seconds--a speedup of more than 3 orders of magnitude.
It is to be understood that the embodiments and variations shown and
described herein are merely illustrative of the principles of this
invention and that various modifications may be implemented by those
skilled in the art without departing from the scope and spirit of the
invention.
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